Wepropose a New Number Representation and Arithmetic For the Elements of the Ringof Integers Modulo P. the So- Called Polynomial Modular Number System (Pmns)Allows For Fast Polynomial Arithmetic and Easy Parallelization. the Mostimportant Contribution of This Paper Is the FundamenTal Theorem of a Modularnumber System, Which Provides a Bound For the Coefficients of the Polynomialsused to RepResent the Set However,We Also Propose a Complete Set of Algorithms to Perform the Arithmeticoperations Over a Pmns, Which Make This System of Practical Interest For Peopleconcerned About Efficient Implementation of ModuLar Arithmetic.